Average Error: 0.2 → 0.0
Time: 12.9s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[6 \cdot \frac{x - 1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
6 \cdot \frac{x - 1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}
double f(double x) {
        double r521952 = 6.0;
        double r521953 = x;
        double r521954 = 1.0;
        double r521955 = r521953 - r521954;
        double r521956 = r521952 * r521955;
        double r521957 = r521953 + r521954;
        double r521958 = 4.0;
        double r521959 = sqrt(r521953);
        double r521960 = r521958 * r521959;
        double r521961 = r521957 + r521960;
        double r521962 = r521956 / r521961;
        return r521962;
}
↓
double f(double x) {
        double r521963 = 6.0;
        double r521964 = x;
        double r521965 = 1.0;
        double r521966 = r521964 - r521965;
        double r521967 = 4.0;
        double r521968 = sqrt(r521964);
        double r521969 = r521964 + r521965;
        double r521970 = fma(r521967, r521968, r521969);
        double r521971 = r521966 / r521970;
        double r521972 = r521963 * r521971;
        return r521972;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
Herbie0.0
\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

Derivation

  1. Initial program 0.2

    \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}}}\]
  3. Using strategy rm

  4. Applied div-inv0.1

    \[\leadsto \color{blue}{6 \cdot \frac{1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}}}\]

  5. Simplified0.0

    \[\leadsto 6 \cdot \color{blue}{\frac{x - 1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}}\]

  6. Final simplification0.0

    \[\leadsto 6 \cdot \frac{x - 1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}\]

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :herbie-target
  (/ 6 (/ (+ (+ x 1) (* 4 (sqrt x))) (- x 1)))

  (/ (* 6 (- x 1)) (+ (+ x 1) (* 4 (sqrt x)))))